Simplify (1/a+1/b)/(((a+b)^2)/(3ab))

$\frac{\frac{1}{a} + \frac{1}{b}}{\frac{{(a + b)}^{2}}{3 a b}}$

Step 1

Multiply the numerator by the reciprocal of the denominator.

$(\frac{1}{a} + \frac{1}{b}) \frac{3 a b}{{(a + b)}^{2}}$

Step 2

To write $\frac{1}{a}$ as a fraction with a common denominator, multiply by $\frac{b}{b}$ .

$(\frac{1}{a} \cdot \frac{b}{b} + \frac{1}{b}) \frac{3 a b}{{(a + b)}^{2}}$

Step 3

To write $\frac{1}{b}$ as a fraction with a common denominator, multiply by $\frac{a}{a}$ .

$(\frac{1}{a} \cdot \frac{b}{b} + \frac{1}{b} \cdot \frac{a}{a}) \frac{3 a b}{{(a + b)}^{2}}$

Step 4

Write each expression with a common denominator of $a b$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{1}{a}$ by $\frac{b}{b}$ .

$(\frac{b}{a b} + \frac{1}{b} \cdot \frac{a}{a}) \frac{3 a b}{{(a + b)}^{2}}$

Multiply $\frac{1}{b}$ by $\frac{a}{a}$ .

$(\frac{b}{a b} + \frac{a}{b a}) \frac{3 a b}{{(a + b)}^{2}}$

Reorder the factors of $b a$ .

$(\frac{b}{a b} + \frac{a}{a b}) \frac{3 a b}{{(a + b)}^{2}}$

Step 5

Combine the numerators over the common denominator.

$\frac{b + a}{a b} \cdot \frac{3 a b}{{(a + b)}^{2}}$

Step 6

Cancel the common factor of $a b$ .

Factor $a b$ out of $3 a b$ .

$\frac{b + a}{a b} \cdot \frac{a b (3)}{{(a + b)}^{2}}$

Cancel the common factor.

$\frac{b + a}{a b} \cdot \frac{a b \cdot 3}{{(a + b)}^{2}}$

Rewrite the expression.

$(b + a) \frac{3}{{(a + b)}^{2}}$

Step 7

Multiply $b + a$ by $\frac{3}{{(a + b)}^{2}}$ .

$\frac{(b + a) \cdot 3}{{(a + b)}^{2}}$

Step 8

Cancel the common factor of $b + a$ and ${(a + b)}^{2}$ .

Reorder terms.

$\frac{(a + b) \cdot 3}{{(a + b)}^{2}}$

Factor $a + b$ out of $(a + b) \cdot 3$ .

$\frac{(a + b) (3)}{{(a + b)}^{2}}$

Cancel the common factors.

Factor $a + b$ out of ${(a + b)}^{2}$ .

$\frac{(a + b) (3)}{(a + b) (a + b)}$

Cancel the common factor.

$\frac{(a + b) \cdot 3}{(a + b) (a + b)}$

Rewrite the expression.

$\frac{3}{a + b}$